Once the tank is full, a roaring or moaning sound comes from the pipes in the wall and basement - any idea what this could be?

If this is a float ball ballcoick I would suspect a loose washer. Shut the water off and remove the three screws on the top of the ballcock.
Pull the float ball and stem assembly up and reseat the washer firmly. Before you reassemble turn the water back on to flush out the supply.
Now reassemble and give it a flush. Good luck, Tom